Subgroups — C₂×Q₈×D₆₀ #14482913

Order 1920.

3136 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₆₀	C₂	1	16	yes	maximal, direct product factor
4	C₂×#91786	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₆₀	C₂	1	3	yes	maximal
6	C₂×#91758	C₂	1	3	yes	maximal, complemented
7	C₂×Q₈×D₂₀	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×D₁₅	—	2	16	no	—
10	C₂×Q₈×D₁₅	C₂²	1	16	yes	complemented
11	C₂²×D₆₀	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₅	C₂²	1	6	yes	—
13	C₂²×Dic₃₀	C₂²	1	6	yes	—
14	C₃×C₄×C₅×Q₈	C₂²	1	8	yes	complemented
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
17	C₂×#6714	C₂²	1	6	yes	—
18	C₂×C₄⋊Dic₁₅	C₂²	1	3	yes	—
19	C₂×C₄⋊Dic₁₅	C₂²	1	6	yes	—
20	#91786	C₂²	1	48	yes	complemented
21	C₄×D₆₀	C₂²	1	24	yes	—
22	#91758	C₂²	1	24	yes	complemented
23	C₂×Q₈×D₁₂	—	5	1	no	maximal, Hall
24	C₂²×Q₈×D₅	—	3	2	no	—
25	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
26	Q₈×D₂₀	—	3	16	no	—
27	C₂×#4043	—	3	6	no	—
28	C₂×C₄×D₂₀	—	3	3	no	—
29	C₂×#4015	—	3	3	no	—
30	C₂³×D₁₅	C₂³	1	2	yes	—
31	C₂×C₃×C₅×Q₈	D₄	1	8	yes	complemented
32	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₅	C₂³	1	6	yes	—
37	Q₈×D₁₅	—	2	64	no	—
38	C₂×D₆₀	Q₈	1	8	yes	direct product factor
39	C₂×D₆₀	C₂³	1	4	yes	—
40	C₂×C₄×D₁₅	—	2	24	no	—
41	C₂×C₄×D₁₅	C₂³	1	24	yes	—
42	C₂×Dic₃₀	—	2	24	no	—
43	C₂×Dic₃₀	C₂³	1	24	yes	—
44	C₃×C₅×C₄⋊C₄	C₂³	1	12	yes	—
45	C₃×C₄²×C₅	C₂³	1	12	yes	—
46	#6714	C₂³	1	24	yes	—
47	C₄⋊Dic₁₅	C₂³	1	12	yes	—
48	C₄⋊Dic₁₅	C₂³	1	24	yes	—
49	C₂²×D₃×Q₈	—	5	2	no	—
50	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
51	Q₈×D₁₂	—	5	16	no	—
52	C₂×#897	—	5	6	no	—
53	C₂×C₄×D₁₂	—	5	3	no	—
54	C₂×#869	—	5	3	no	—
55	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
56	C₂×Q₈×D₅	—	3	16	no	—
57	C₂×Q₈×D₅	—	6	16	no	—
58	C₂²×D₂₀	—	3	1	no	—
59	C₂²×C₄×D₅	—	3	6	no	—
60	C₂²×Dic₁₀	—	3	6	no	—
61	C₄×C₅×Q₈	C₂×D₃	1	8	yes	complemented
62	C₂×C₅×C₄⋊C₄	C₂×D₃	1	3	yes	—
63	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
64	C₂×#685	—	3	6	no	—
65	C₂×C₄⋊Dic₅	—	3	3	no	—
66	C₂×C₄⋊Dic₅	—	3	6	no	—
67	#4043	—	3	48	no	—
68	C₄×D₂₀	—	3	24	no	—
69	#4015	—	3	24	no	—
70	C₂×D₄×Q₈	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₂²×D₁₅	—	2	4	no	—
73	C₂²×D₁₅	—	2	4	no	—
74	C₂²×D₁₅	C₂×Q₈	1	8	yes	complemented
75	C₂²×D₁₅	C₂⁴	1	4	yes	—
76	C₃×C₅×Q₈	C₂×D₄	1	16	yes	complemented
77	C₂×C₃×C₄×C₅	—	2	6	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
79	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	complemented
80	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
83	C₂×Dic₁₅	—	2	12	no	—
84	C₂×Dic₁₅	C₂⁴	1	12	yes	—
85	D₆₀	C₂×Q₈	1	16	yes	direct product factor
86	C₄×D₁₅	—	2	48	no	—
87	Dic₃₀	—	2	48	no	—
88	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
89	C₂×D₃×Q₈	—	5	16	no	—
90	C₂×D₃×Q₈	—	10	16	no	—
91	C₂²×D₁₂	—	5	1	no	—
92	C₂²×C₄×D₃	—	5	6	no	—
93	C₂²×Dic₆	—	5	6	no	—
94	C₃×C₄×Q₈	C₂×D₅	1	8	yes	complemented
95	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
96	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
97	C₂×(C₂×D₃)⋊C₄	—	5	6	no	—
98	C₂×C₄⋊Dic₃	—	5	3	no	—
99	C₂×C₄⋊Dic₃	—	5	6	no	—
100	#897	—	5	48	no	—
101	C₄×D₁₂	—	5	24	no	—
102	#869	—	5	24	no	—
103	C₂³×D₅	—	3	2	no	—
104	C₂×C₅×Q₈	D₁₂	1	8	yes	complemented
105	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
106	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
107	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂²×Dic₅	—	3	6	no	—
110	Q₈×D₅	—	6	64	no	—
111	C₂×D₂₀	—	3	8	no	—
112	C₂×D₂₀	—	3	4	no	—
113	C₂×C₄×D₅	—	3	24	no	—
114	C₂×C₄×D₅	—	6	24	no	—
115	C₂×Dic₁₀	—	3	24	no	—
116	C₂×Dic₁₀	—	6	24	no	—
117	C₅×C₄⋊C₄	C₂²×D₃	1	12	yes	—
118	C₄²×C₅	C₂²×D₃	1	12	yes	—
119	#685	—	3	24	no	—
120	C₄⋊Dic₅	—	3	12	no	—
121	C₄⋊Dic₅	—	3	24	no	—
122	C₂³×Q₈	—	15	2	no	—
123	D₄×Q₈	—	15	16	no	—
124	C₂×C₄⋊Q₈	—	15	3	no	—
125	C₂×C₂²⋊Q₈	—	15	6	no	—
126	C₂×C₄×Q₈	D₁₅	1	1	yes	characteristic, complemented
127	C₂×C₄×D₄	—	15	3	no	—
128	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
129	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
130	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
131	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₅	—	2	16	no	—
133	C₂×D₁₅	—	2	4	no	—
134	C₂×D₁₅	—	2	4	no	—
135	C₂×D₁₅	C₂²×Q₈	1	8	yes	complemented
136	C₃×C₄×C₅	—	2	6	no	—
137	C₃×C₄×C₅	C₂²×D₄	1	12	yes	—
138	C₃×C₄×C₅	C₂²×Q₈	1	4	yes	complemented
139	Dic₁₅	—	2	12	no	—
140	C₂³×D₃	—	5	2	no	—
141	C₂×C₃×Q₈	D₂₀	1	8	yes	complemented
142	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
143	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
144	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
145	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
146	C₂²×Dic₃	—	5	6	no	—
147	D₃×Q₈	—	10	64	no	—
148	C₂×D₁₂	—	5	8	no	—
149	C₂×D₁₂	—	5	4	no	—
150	C₂×C₄×D₃	—	5	24	no	—
151	C₂×C₄×D₃	—	10	24	no	—
152	C₂×Dic₆	—	5	24	no	—
153	C₂×Dic₆	—	10	24	no	—
154	C₃×C₄⋊C₄	C₂²×D₅	1	12	yes	—
155	C₃×C₄²	C₂²×D₅	1	12	yes	—
156	(C₂×D₃)⋊C₄	—	5	24	no	—
157	C₄⋊Dic₃	—	5	12	no	—
158	C₄⋊Dic₃	—	5	24	no	—
159	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
160	C₂²×D₅	—	3	8	no	—
161	C₂²×D₅	—	3	4	no	—
162	C₂²×D₅	—	6	4	no	—
163	C₂²×D₅	—	6	4	no	—
164	C₅×Q₈	C₂×D₁₂	1	16	yes	complemented
165	C₂×C₄×C₅	—	2	6	no	—
166	C₂×C₄×C₅	C₂×D₁₂	1	12	yes	—
167	C₂×C₄×C₅	D₃×Q₈	1	4	yes	complemented
168	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
169	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
170	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
171	C₂×Dic₅	—	3	12	no	—
172	C₂×Dic₅	—	6	12	no	—
173	D₂₀	—	3	16	no	—
174	C₄×D₅	—	6	48	no	—
175	Dic₁₀	—	6	48	no	—
176	C₂²×Q₈	—	15	16	no	—
177	C₂²×Q₈	—	15	6	no	—
178	C₂²×Q₈	—	30	16	no	—
179	C₂²×Q₈	C₂×D₁₅	1	1	yes	characteristic
180	C₂²×D₄	—	15	1	no	—
181	C₂³×C₄	—	15	6	no	—
182	C₄⋊Q₈	—	15	24	no	—
183	C₂²⋊Q₈	—	15	48	no	—
184	C₄×Q₈	C₂×D₁₅	1	8	yes	complemented
185	C₄×D₄	—	15	24	no	—
186	C₂×C₄⋊C₄	—	15	3	no	—
187	C₂×C₄⋊C₄	—	15	6	no	—
188	C₂×C₄⋊C₄	C₂×D₁₅	1	3	yes	—
189	C₂×C₂²⋊C₄	—	15	6	no	—
190	C₂×C₄²	C₂×D₁₅	1	3	yes	—
191	C₂×C₃×C₅	D₄×Q₈	1	4	yes	complemented
192	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
195	D₁₅	—	2	8	no	—
196	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
197	C₂²×D₃	—	5	8	no	—
198	C₂²×D₃	—	5	4	no	—
199	C₂²×D₃	—	10	4	no	—
200	C₂²×D₃	—	10	4	no	—
201	C₃×Q₈	C₂×D₂₀	1	16	yes	complemented
202	C₂×C₃×C₄	—	2	6	no	—
203	C₂×C₃×C₄	C₂×D₂₀	1	12	yes	—
204	C₂×C₃×C₄	Q₈×D₅	1	4	yes	complemented
205	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
206	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
207	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
208	C₂×Dic₃	—	5	12	no	—
209	C₂×Dic₃	—	10	12	no	—
210	D₁₂	—	5	16	no	—
211	C₄×D₃	—	10	48	no	—
212	Dic₆	—	10	48	no	—
213	C₂²×C₅	C₂²×D₁₂	1	2	yes	—
214	C₂²×C₅	C₂×D₃×Q₈	1	2	yes	—
215	C₂²×C₅	#1010	1	2	yes	—
216	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
217	C₂×D₅	—	3	8	no	—
218	C₂×D₅	—	6	4	no	—
219	C₂×D₅	—	6	4	no	—
220	C₂×D₅	—	6	16	no	—
221	C₄×C₅	—	2	6	no	—
222	C₄×C₅	C₂²×D₁₂	1	12	yes	—
223	C₄×C₅	C₂×D₃×Q₈	1	4	yes	complemented
224	Dic₅	—	6	12	no	—
225	C₂⁴	—	15	2	no	—
226	C₂×Q₈	—	15	24	no	—
227	C₂×Q₈	—	30	64	no	—
228	C₂×Q₈	—	30	24	no	—
229	C₂×Q₈	D₆₀	1	8	yes	direct product factor
230	C₂×Q₈	C₂²×D₁₅	1	4	yes	—
231	C₂×D₄	—	15	8	no	—
232	C₂×D₄	—	15	4	no	—
233	C₂²×C₄	—	15	6	no	—
234	C₂²×C₄	—	15	24	no	—
235	C₂²×C₄	—	30	24	no	—
236	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
237	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
238	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
239	C₄⋊C₄	—	15	12	no	—
240	C₄⋊C₄	—	15	24	no	—
241	C₄⋊C₄	C₂²×D₁₅	1	12	yes	—
242	C₂²⋊C₄	—	15	24	no	—
243	C₄²	C₂²×D₁₅	1	12	yes	—
244	C₃×C₅	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	C₂²×D₂₀	1	2	yes	—
246	C₂²×C₃	C₂×Q₈×D₅	1	2	yes	—
247	C₂²×C₃	#4150	1	2	yes	—
248	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
249	C₂×D₃	—	5	8	no	—
250	C₂×D₃	—	10	16	no	—
251	C₂×D₃	—	10	4	no	—
252	C₂×D₃	—	10	4	no	—
253	C₃×C₄	—	2	6	no	—
254	C₃×C₄	C₂²×D₂₀	1	12	yes	—
255	C₃×C₄	C₂×Q₈×D₅	1	4	yes	complemented
256	Dic₃	—	10	12	no	—
257	C₂×C₅	Q₈×D₁₂	1	4	yes	complemented
258	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
259	C₂×C₅	C₂²×D₃×Q₈	1	1	yes	characteristic
260	C₂×C₅	C₂×#1010	1	1	yes	characteristic
261	D₅	—	6	8	no	—
262	C₂³	—	15	8	no	—
263	C₂³	—	15	4	no	—
264	C₂³	—	30	4	no	—
265	C₂³	—	30	4	no	—
266	C₂³	C₂³×D₁₅	1	1	yes	characteristic
267	Q₈	—	30	48	no	—
268	Q₈	C₂×D₆₀	1	16	yes	direct product factor
269	D₄	—	15	16	no	—
270	C₂×C₄	—	2	6	no	—
271	C₂×C₄	—	15	12	no	—
272	C₂×C₄	—	30	12	no	—
273	C₂×C₄	—	30	48	no	—
274	C₂×C₄	C₂×D₆₀	1	12	yes	—
275	C₂×C₄	Q₈×D₁₅	1	4	yes	complemented
276	C₂×C₄	C₂³×D₁₅	1	6	yes	—
277	C₂×C₄	C₂³×D₁₅	1	6	yes	—
278	C₂×C₄	C₂³×D₁₅	1	2	yes	—
279	C₂×C₃	Q₈×D₂₀	1	4	yes	complemented
280	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
281	C₂×C₃	C₂²×Q₈×D₅	1	1	yes	characteristic
282	C₂×C₃	C₂×#4150	1	1	yes	characteristic
283	D₃	—	10	8	no	—
284	C₅	C₂×Q₈×D₁₂	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	15	8	no	—
286	C₂²	—	30	16	no	—
287	C₂²	—	30	4	no	—
288	C₂²	—	30	4	no	—
289	C₂²	C₂²×D₆₀	1	2	yes	—
290	C₂²	C₂×Q₈×D₁₅	1	2	yes	—
291	C₂²	#92101	1	2	yes	—
292	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
293	C₄	—	2	6	no	—
294	C₄	—	30	12	no	—
295	C₄	C₂²×D₆₀	1	12	yes	—
296	C₄	C₂×Q₈×D₁₅	1	4	yes	complemented
297	C₃	C₂×Q₈×D₂₀	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	30	8	no	—
299	C₂	Q₈×D₆₀	1	4	yes	direct product factor
300	C₂	C₂³×D₆₀	1	1	yes	characteristic
301	C₂	C₂²×Q₈×D₁₅	1	1	yes	characteristic
302	C₂	C₂×#92101	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×D₆₀	C₂²
hypercenter	C₂×C₄×Q₈	D₁₅
hypocenter	C₃×C₅	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₅	C₂	semidirect
2. C₂×C₃×C₄×C₅×Q₈	C₂	semidirect
3. Q₈×D₆₀	C₂	direct (N × H)
4. C₂×#91786	C₂	semidirect
6. C₂×#91758	C₂	semidirect
10. C₂×Q₈×D₁₅	C₂²	semidirect
14. C₃×C₄×C₅×Q₈	C₂²	semidirect
20. #91786	C₂²	semidirect
22. #91758	C₂²	semidirect
25. C₂×C₄×C₅×Q₈	D₃	semidirect
31. C₂×C₃×C₅×Q₈	D₄	semidirect
38. C₂×D₆₀	Q₈	direct (N × H)
50. C₂×C₃×C₄×Q₈	D₅	semidirect
61. C₄×C₅×Q₈	C₂×D₃	semidirect
74. C₂²×D₁₅	C₂×Q₈	semidirect
76. C₃×C₅×Q₈	C₂×D₄	semidirect
79. C₂×C₃×C₄×C₅	C₂×Q₈	semidirect
85. D₆₀	C₂×Q₈	direct (N × H)
94. C₃×C₄×Q₈	C₂×D₅	semidirect
104. C₂×C₅×Q₈	D₁₂	semidirect
126. C₂×C₄×Q₈	D₁₅	semidirect
135. C₂×D₁₅	C₂²×Q₈	semidirect
138. C₃×C₄×C₅	C₂²×Q₈	semidirect
141. C₂×C₃×Q₈	D₂₀	semidirect
164. C₅×Q₈	C₂×D₁₂	semidirect
167. C₂×C₄×C₅	D₃×Q₈	semidirect
184. C₄×Q₈	C₂×D₁₅	semidirect
191. C₂×C₃×C₅	D₄×Q₈	semidirect
201. C₃×Q₈	C₂×D₂₀	semidirect
204. C₂×C₃×C₄	Q₈×D₅	semidirect
223. C₄×C₅	C₂×D₃×Q₈	semidirect
229. C₂×Q₈	D₆₀	direct (N × H)
244. C₃×C₅	C₂×D₄×Q₈	semidirect
255. C₃×C₄	C₂×Q₈×D₅	semidirect
257. C₂×C₅	Q₈×D₁₂	semidirect
268. Q₈	C₂×D₆₀	direct (N × H)
275. C₂×C₄	Q₈×D₁₅	semidirect
279. C₂×C₃	Q₈×D₂₀	semidirect
284. C₅	C₂×Q₈×D₁₂	semidirect
296. C₄	C₂×Q₈×D₁₅	semidirect
297. C₃	C₂×Q₈×D₂₀	semidirect
299. C₂	Q₈×D₆₀	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Q₈×D₁₅	2
C₂×C₃×C₄×C₅×Q₈	2
Q₈×D₆₀	2
C₂×#91786	2
C₂×C₄×D₆₀	2
C₂×#91758	2
C₂×Q₈×D₂₀	3
C₂×Q₈×D₁₂	5

Subgroups — C2×Q8×D60 #14482913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₆₀ #14482913